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## Inventory Control

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**Inventory Control**IME 451, Lecture 3**Economic Order Quantity**• Harris (1913) developed this basic, widely used model to find economic lot sizes • Balancing inventory holding costs against setup (or order) costs • Assumptions • Production is instantaneous, no capacity constraint • Delivery is immediate, no time lag • Demand is deterministic, no uncertainty • Demand is constant over time • A production run incurs a fixed setup cost • Products can be analyzed individually (single product only or no interactions such as shared resources or machines)**EOQ Variables**• D = demand rate (in units per year) • c = unit production cost, not counting setup or inventory cost (in dollars per unit) • A = fixed setup (ordering) cost to produce (purchase) a lot (in dollars) • h = holding cost (in dollars per unit per year); if holding costs are only due to interest then h=ic where i is the annual interest rate • Q = lot size (in units); this is the decision variable**EOQ Derivation**• Total annual cost Y(Q) • Find the minimum Y(Q) by setting derivative w.r.t. Q equal to 0 • Check that the second derivative is positive for any positive Q (convex function, so Q* is a min, not a max) • Solve first derivative for Q***Problems with EOQ**• How realistic are assumptions (Instantaneous production? Deterministic demand?) • Setup costs may be difficult to estimate, especially in production environments rather than purchasing systems • Average number of lots per year, F • Total inventory investment, I • Time between orders, T**Sensitivity of EOQ Models**• Holding and setup costs are fairly insensitive to lot size • Errors caused by restricting lot sizes to powers of 2 are minimal (no more than 6%) • Powers of 2 ordering can facilitate sharing truck resources (one week, two weeks, four weeks…) • Extensions involve non-instantaneous production (economic production lot model), backorders, major and minor setups, and quantity discounts**Dynamic Lot Sizing (Wagner Whitin)**• Notation t = a time period = 1, 2, …, T where T is the planning horizon Dt = demand in period t (in units) ct = unit production cost (in dollars per unit) At = setup cost to produce a lot in period t (in dollars) ht = holding cost to carry a unit of inventory from period t to period t+1 (in dollars per unit per period) It = inventory (in units) leftover at the end of period t Qt = lot size for period t (in units); there are T decision variables, one for each period**Wagner Whitin Procedure**• Qt will be 0 or will be Dt, Dt+Dt+1 , Dt+Dt+1+Dt+2 ... • Produce nothing or produce exactly enough to cover the current period plus some integer number of future periods • Produce for the first period in the first period • For each subsequent period, decide whether it is more economical to produce that period’s demand in the current period or any previous period • Follow example in book, pp. 60-61**Models for Uncertain Demand**• Finding statistical reorder points to account for randomness in demand • News Vendor – single replenishment; vendor buys paper at start of day and discards any leftover at end of day • Base Stock – replenish inventory one unit at a time but carry base stock to cover lag time • (Q,r) model – when inventory reaches or falls below level r, order a quantity of Q items**News Vendor Model**• Assumptions • Products are separable, consider 1 at a time • Planning is for a single period only • Demand is random • Deliveries are made in advance of demand • Costs of overage or underage are linear • Notation • X = demand (in units), a random variable with mean m and standard deviation s • G(x) = P(X<=x) = c.d.f. of demand • cs = cost (in dollars) per unit of shortage • co = cost (in dollars) per unit of overage • Q = production or order quantity (in units); this is the decision variable**News Vendor Equations**• To balance overage vs shortage costs, choose order quantity Q* • Assume that G is normal, where F is the cdf of the standard normal function • Find z in a standard normal table (Table 1 at end of text) • Solve for Q***Base Stock Model**• Assumptions • Products can be analyzed individually • Demands occur one at a time • Unfilled demand is backordered • Replenishment leadtimes are fixed and known • Replenishments are ordered one at a time • Notation • l = replenishment leadtime (in days), assumed constant • X = demand during replenishment leadtime (in units) • G(x) = P(X<=x) = c.d.f. probability demand during replenishment leadtime is less than or equal to x • q = E[X] = mean demand (in units) during leadtime l • s = E[X] = standard deviation of demand (in units) during leadtime l**Base Stock Model**• Notation (continued) • h = cost to carry one unit of inventory for one year • b = cost to carry one unit of backorder for one year • r = reorder point (in units) • R = r + 1 = base stock level (in units) • s = r - q = safety stock level (in units) • S( R ) = fill rate or service level, fraction of orders filled from stock as a function of R • B( R ) = average outstanding backorders • I( R ) = average on-hand inventory • Inventory position = on-hand inventory – backorders + orders**Base Stock Equations**• For this model, at all times inventory position = R • Service level, S( R ) = G (R – 1) • Backorders are 0 if x < R • Backorders are x-R if x>=R • Expected backorder level B( R ) • I ( R ) = R – q + B( R ) • Use table 2.5 to find fill rates**(Q,r) Model**• Assumptions • Similar to base stock except • There is a fixed cost for each replenishment order, OR • There is a constraint on the number of orders per year • Decide how much safety stock to carry to cover leadtimes AND what quantity to order**(Q,r) Notation**• D = expected demand per year (in units) • l = replenishment leadtime (in days) • X = demand during replenishment leadtime (in units), random variable • q = E[X] = Dl/365 = expected demand (in units) during leadtime l • s = E[X] = standard deviation of demand (in units) during leadtime l • G(x) = P(X<=x) = c.d.f. probability demand during replenishment leadtime is less than or equal to x • A = setup cost per replenishment (in dollars) • c = unit production cost (in dollars per unit) • h = cost to carry one unit of inventory for one year • k = cost per stockout • b = cost to carry one unit of backorder for one year • r = reorder point (in units) • Q = replenishment quantity • s = r - q = safety stock level (in units) • F (Q,r) = order frequency, as a fuction of Q and r • S( Q,r ) = fill rate or service level, fraction of orders filled from stock as a function of Q and r • B( Q,r ) = average outstanding backorders • I( Q,r ) = average on-hand inventory**(Q,r) Equations**• Replenishment quantity Q affects cycle stock, inventory that is held to avoid excessive replenishment costs (like EOQ) • Reorder point r affects safety stock, inventory held to avoid stockouts (like Base Stock) • Either minimize: or**Fixed Setup Cost & Backorder Cost**• Fixed setup (or order) cost • First, set number of orders per year • Then, find annual fixed order costs • Backorder Cost • Inventory position is uniformly distributed between r+1 and r+Q • Averaging the backorder level over the range r+1 to r+Q**Stockout Cost in (Q,r) Model**• Penalizes poor customer service • Charge a cost each time a stockout occurs • Charge a penalty that is proportional to the time a customer waits to have their order filled • Approximations • Type I (base stock) – computes # of stockouts per cycle, underestimates S(Q,r) • Type II – neglects B(Q,r) term, also underestimates S(Q,r)**Holding Costs in (Q,r) Model**• Inventory holding cost = hI(Q,r) • First equation approximates and underestimates average inventory, since demand is variable and thus backorders sometimes occur • Exact formulation:**Backorder Cost Approach**• Compute Q and r values that minimize: • approximate, since B(r) replaces B(Q,r) • Optimal Q* and r* • Then, assume normal distribution for lead time as in base stock model**Stockout Cost Approach**• Compute Q and r values that minimize: • approximate, since B(r) replaces B(Q,r) • Optimal Q* and r* • Assume normal distribution for lead time • Note: larger Q results in smaller r* because a smaller reorder point is needed to achieve the same fill rate**(Q,r) Insights**• Cycle stock increases as replenishment frequency decreases • Safety stock provides a buffer against stockouts • The base stock model is a (Q,r) model where Q=1 • Increasing annual demand (D) increases Q • Increasing demand during leadtime increases r (thus, high annual demand or long leadtimes require inventory) • Increased demand variability increases r (for most parts where high fill rates are desired) • Increased holding costs (h) decreases both Q and r (if it is expensive to hold inventory, avoid doing so!)